summary - ucsbmotion.me.ucsb.edu/talks/2010u-synchkron-annecy-13sep10-2x2.pdf · er and bullo...

Synchronization and Kron Reduction in Power Networks

Florian Dorfler and Francesco Bullo

Center for Control,Dynamical Systems & Computation

University of California at Santa Barbara

IFAC Workshop on Distributed Estimation and Control in Networked SystemsAnnecy, France, 13-14 September, 2010

Poster tomorrow on: “Network reduction and effective resistance”Slides and papers available at: http://motion.me.ucsb.edu

Dorfler and Bullo (UCSB) Synchronization & Kron Reduction NECSYS ’10 @ Annecy 1 / 30

Summary

observations from distinct fields:

1 power networks are coupled oscillators

2 Kuramoto oscillators synchronize for large coupling

3 graph theory quantifies coupling in a network

4 hence, power networks synchronize for large coupling

Today’s talk:

theorems about these observations

synch tests for “net-preserving” and “reduced” models


Outline

1 IntroductionMotivationMathematical modelProblem statement

2 Singular perturbation analysis(to relate power network and Kuramoto model)

3 Synchronization of non-uniform Kuramoto oscillators

4 Network-preserving power network models

5 Conclusions


Motivation: the current US power grid

“. . . the largest and most complex machine engineered by humankind.”

[P. Kundur ’94, V. Vittal ’03, . . . ]

“. . . the greatest engineering achievement of the 20th century.”

[National Academy of Engineering ’10]

1 large-scale, nonlinear dynamics, complex interactions

2 100 years old and operating at its capacity limits

⇒ recent blackouts: New England ’03, Italy ’03, Brazil ’09


Motivation: the future smart grid

Energy is one of the top three national priorities, [B. Obama, ’09]

Expected developments in “smart grid”:

⇒ increasing consumption

⇒ increasing adoption of renewable power sources:1 large number of distributed power sources2 power transmission from remote areas

⇒ large-scale heterogeneous networks with stochastic disturbances

Transient Stability

Generators to swing synchronously despitevariability/faults in generators/network/loads


Motivation: the Mediterranean ring project

Synchronous grid interconnection between EU and Mediterranean region:

1 Provide increased levels of energy security to participating nations;

2 Import/export electric power among nations;

3 Cut back on the primary electricity reserve requirements within each country.

Reference: “Oscillation behavior of the enlarged European power system” by M. Kurthand E. Welfonder. Control Engineering Practice, 2005.


Mathematical model of a power network

New England Power Grid

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F

Fig. 9. The New England test system [10], [11]. The system includes10 synchronous generators and 39 buses. Most of the buses have constantactive and reactive power loads. Coupled swing dynamics of 10 generatorsare studied in the case that a line-to-ground fault occurs at point F near bus16.

test system can be represented by

!i = "i,Hi

#fs

"i = !Di"i + Pmi !GiiE2i !

10!

j=1,j !=i

EiEj ·

· {Gij cos(!i ! !j) + Bij sin(!i ! !j)},

"

#

#

$

#

#

%

(11)

where i = 2, . . . , 10. !i is the rotor angle of generator i withrespect to bus 1, and "i the rotor speed deviation of generatori relative to system angular frequency (2#fs = 2#" 60Hz).!1 is constant for the above assumption. The parametersfs, Hi, Pmi, Di, Ei, Gii, Gij , and Bij are in per unitsystem except for Hi and Di in second, and for fs in Helz.The mechanical input power Pmi to generator i and themagnitude Ei of internal voltage in generator i are assumedto be constant for transient stability studies [1], [2]. Hi isthe inertia constant of generator i, Di its damping coefficient,and they are constant. Gii is the internal conductance, andGij + jBij the transfer impedance between generators iand j; They are the parameters which change with networktopology changes. Note that electrical loads in the test systemare modeled as passive impedance [11].

B. Numerical Experiment

Coupled swing dynamics of 10 generators in thetest system are simulated. Ei and the initial condition(!i(0),"i(0) = 0) for generator i are fixed through powerflow calculation. Hi is fixed at the original values in [11].Pmi and constant power loads are assumed to be 50% at theirratings [22]. The damping Di is 0.005 s for all generators.Gii, Gij , and Bij are also based on the original line datain [11] and the power flow calculation. It is assumed thatthe test system is in a steady operating condition at t = 0 s,that a line-to-ground fault occurs at point F near bus 16 att = 1 s!20/(60Hz), and that line 16–17 trips at t = 1 s. Thefault duration is 20 cycles of a 60-Hz sine wave. The faultis simulated by adding a small impedance (10"7j) betweenbus 16 and ground. Fig. 10 shows coupled swings of rotorangle !i in the test system. The figure indicates that all rotorangles start to grow coherently at about 8 s. The coherentgrowing is global instability.

C. Remarks

It was confirmed that the system (11) in the New Eng-land test system shows global instability. A few comments

0 2 4 6 8 10-5

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Fig. 10. Coupled swing of phase angle !i in New England test system.The fault duration is 20 cycles of a 60-Hz sine wave. The result is obtainedby numerical integration of eqs. (11).

are provided to discuss whether the instability in Fig. 10occurs in the corresponding real power system. First, theclassical model with constant voltage behind impedance isused for first swing criterion of transient stability [1]. This isbecause second and multi swings may be affected by voltagefluctuations, damping effects, controllers such as AVR, PSS,and governor. Second, the fault durations, which we fixed at20 cycles, are normally less than 10 cycles. Last, the loadcondition used above is different from the original one in[11]. We cannot hence argue that global instability occurs inthe real system. Analysis, however, does show a possibilityof global instability in real power systems.

IV. TOWARDS A CONTROL FOR GLOBAL SWING

INSTABILITY

Global instability is related to the undesirable phenomenonthat should be avoided by control. We introduce a keymechanism for the control problem and discuss controlstrategies for preventing or avoiding the instability.

A. Internal Resonance as Another Mechanism

Inspired by [12], we here describe the global instabilitywith dynamical systems theory close to internal resonance[23], [24]. Consider collective dynamics in the system (5).For the system (5) with small parameters pm and b, the set{(!,") # S1 " R | " = 0} of states in the phase plane iscalled resonant surface [23], and its neighborhood resonantband. The phase plane is decomposed into the two parts:resonant band and high-energy zone outside of it. Here theinitial conditions of local and mode disturbances in Sec. IIindeed exist inside the resonant band. The collective motionbefore the onset of coherent growing is trapped near theresonant band. On the other hand, after the coherent growing,it escapes from the resonant band as shown in Figs. 3(b),4(b), 5, and 8(b) and (c). The trapped motion is almostintegrable and is regarded as a captured state in resonance[23]. At a moment, the integrable motion may be interruptedby small kicks that happen during the resonant band. That is,the so-called release from resonance [23] happens, and thecollective motion crosses the homoclinic orbit in Figs. 3(b),4(b), 5, and 8(b) and (c), and hence it goes away fromthe resonant band. It is therefore said that global instability

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Authorized licensed use limited to: Univ of Calif Santa Barbara. Downloaded on June 10, 2009 at 14:48 from IEEE Xplore. Restrictions apply.

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Power network topology:

1 n generators �� , each connected to a generator terminal bus �♦

2 n generators terminal buses �♦ and m load buses •◦ form connected graph

3 admittance matrix Ynetwork∈ C(2n+m)×(2n+m) characterizes the network



Network-preserving DAE power network model:

1 n generators �� = boundary nodes:

Mi

πf0θi = −Di θi + Pmech.in,i − Pelectr.out,i

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2 n + m passive �♦ & •◦ = interior nodes:

• loads are modeled as shunt admittances

• algebraic Kirchhoff equations:

I = YnetworkV

Y1,3Y1,2

Y1,shunt

1



Network-Reduction to an ODE power network model

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[Iboundary

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[Yboundary Ybound-int

Y Tbound-int Yinterior

]︸︷︷︸

Ynetwork

[Vboundary

Vinterior

]

Schur complement

Yreduced = Ynetwork/Yinterior =⇒ Iboundary = YreducedVboundary

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network reduced to active nodes (generators)

Yreduced induces complete “all-to-all” coupling graph



Network-Reduced ODE power network model:

classic interconnected swing equations[Anderson et al. ’77, M. Pai ’89, P. Kundur ’94, . . . ]:

Mi

πf0θi = −Di θi + ωi −

∑j 6=i

Pij sin(θi − θj + ϕij)

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“all-to-all” reduced network Yreduced with

Pij = |Vi ||Vj | |Yreduced,i , j | > 0 max. power transferred i ↔ j

ϕij = arctan(<(Yred,i , j)/=(Yred,i , j)) ∈ [0, π/2) reflect losses i ↔ j

ωi = Pmech.in,i − |Vi |2<(Yreduced,i , i ) effective power input of i


Transient stability analysis: problem statement

Mi


∑j 6=i


Classic transient stability:

1 power network in stable frequency equilibrium(θi , θi ) = (0, 0) for all i

2 → transient network disturbance and fault clearance

3 stability analysis of a new frequency equilibrium in post-fault network

General synchronization problem:

• synchronous equilibrium: |θi − θj | small & θi = θj for all i , j


Transient stability analysis: literature review

Mi


∑j 6=i


Classic methods use Hamiltonian and gradient systems arguments:

1 writeMi

πf0θi = −Di θi −∇iU(θ)T

2 study θi = −∇iU(θ)T

Key objective: compute domain of attraction via numerical methods[N. Kakimoto et al. ’78, H.-D. Chiang et al. ’94 ]

Open Problem “power sys dynamics + complex nets” [Hill and Chen ’06]

transient stability, performance, and robustness of a power network?

! underlying graph properties (topological, algebraic, spectral, etc)


Detour: consensus protocols & Kuramoto oscillators

Consensus protocol in Rn:

xi = −∑

j 6=iaij(xi − xj)

n identical agents with statevariable xi ∈ Rapplication: agreement andcoordination algorithms, . . .

references: [M. DeGroot ’74,

J. Tsitsiklis ’84, L. Moreau ’04, ...]

R

Kuramoto model in Tn:

θi = ωi −K

n

∑j 6=i

sin(θi − θj)

n non-identical oscillators withphase θi ∈ T & frequency ωi ∈ Rapplication: sync phenomena innature, Josephson junctions, . . .

references: [C. Huygens XVII,

Y. Kuramoto ’75, A. Winfree ’80, ...]

T


Detour: synchrony among oscillators

Kuramoto model in Tn

θi = ωi −K

n

∑j 6=i

sin(θi − θj)

notions of synchronization

1 phase cohesiveness: |θi (t)− θj(t)| < γfor small γ < π/2 ... arc invariance

2 frequency synchronized: θi (t) = θj(t)

3 phase synchronized: θi (t) = θj(t)

Detour – Kron reduction of graphs

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Some properties of the Kron reduction process:

1 Well-posedness: Symmetric & irreducible (loopy) Laplacian matricescan be reduced and are closed under Kron reduction

2 Topological properties:

interior network connected ! reduced network complete

at least one node in interior network features a self-loop !

! all nodes in reduced network feature self-loops !

3 Algebraic properties: self-loops in interior network . . .

decrease mutual coupling in reduced network

increase self-loops in reduced network

Florian Dorfler (UCSB) Power Networks Synchronization Advancement to Candidacy 31 / 36


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Classic intuition:

1 K small & |ωi − ωj | large ⇒ no synchronization

2 K large & |ωi − ωj | small ⇒ cohesive + freq synchronization


The big picture

Mi

!f0"i = !Di"i + #i !

!j !=i

Pij sin("i ! "j + $ij)

Consensus Protocols:

xi = !!

j !=iaij(xi ! xj)

Kuramoto Oscillators:

!i = "i !K

n

!j !=i

sin(!i ! !j)

?

Open problem in synchronization and transient stability in power networks:relation to underlying network state, parameters, and topology


The big picture

Mi

!f0"i = !Di"i + #i !

!j !=i

Pij sin("i ! "j + $ij)

Consensus Protocols:

xi = !!

j !=iaij(xi ! xj)

Kuramoto Oscillators:

!i = "i !K

n

!j !=i

sin(!i ! !j)

?


Previous observations about this connection:Power systems: [D. Subbarao et al., ’01, G. Filatrella et al., ’08, V. Fioriti et al., ’09]Networked control: [D. Hill et al., ’06, M. Arcak, ’07]Dynamical systems: [H. Tanaka et al., ’97, A. Arenas ’08]


Outline





5 Conclusions


Singular perturbation analysis

Mi


∑j 6=i


1 assume time-scale separation between synchronization and damping

singular perturbation parameter ε =Mmax

πf0Dmin

2 non-uniform Kuramoto (slow time-scale, for ε = 0)

Di θi = ωi −∑

j 6=iPij sin(θi − θj + ϕij)

3 if cohesiveness + exponential freq sync for non-uniform Kuramoto,then ∀ (θ(0), θ(0)), exists ε∗ > 0 such that ∀ ε < ε∗ and ∀ t ≥ 0

θi (t)power network − θi (t)non-uniform Kuramoto = O(ε)


Singular perturbation analysis: proof

Key technical problem:

Kuramoto defined over manifold Tn, no fixed point

Tikhonov’s Theorem: exp. stable point in Euclidean space

Solution

define grounded variables in Rn−1

δ1 = θ1 − θn · · · δn−1 = θn−1 − θn

equivalence of solutions:1 grounded Kuramoto solutions satisfy maxi,j(δi (t)− δi (t)) < π2 Kuramoto solutions are arc invariant with γ = π,

ie, θ1(t), . . . , θn(t) belong to open half-circle, function of t

equivalence of exponential convergence1 exponential frequency synchronization for Kuramoto2 exponential convergence to equilibrium for grounded Kuramoto


Singular perturbation analysis: discussion

assumption ε =Mmax

πf0Dminsufficiently small

1 generator internal control effects imply ε ∈ O(0.1)

2 topological equivalence independent of ε: 1st-order and 2nd-ordermodels have the same equilibria, the Jacobians have the same inertia,and the regions of attractions are bounded by the same separatrices

3 non-uniform Kuramoto corresponds to reduced gradient systemθi = −∇iU(θ)T used successfully in academia and industry since 1978

4 physical interpretation: damping and sync on separate time-scales

5 classic assumption in literature on coupled oscillators: over-dampedmechanical pendula and Josephson junctions

6 simulation studies show accurate approximation even for large ε


Outline





5 Conclusions


Synchronization of non-uniform Kuramoto: condition

Non-uniform Kuramoto Model in Tn:

Di θi = ωi −∑


Non-uniformity in network: Di , ωi , Pij , ϕij

Phase shift ϕij induces lossless and lossy coupling:

θi =ωi

Di−∑

j 6=i

(Pij

Dicos(ϕij) sin(θi − θj) +

Pij

Disin(ϕij) cos(θi − θj)

)

Synchronization condition (?)

nPmin

Dmaxcos(ϕmax)︸︷︷︸

worst lossless coupling

> maxi ,j

(ωi

Di−

ωj

Dj

)︸︷︷︸

worst non-uniformity

+ maxi

∑j

Pij

Disin(ϕij)︸︷︷︸

worst lossy coupling


Synchronization of non-uniform Kuramoto: consequences

Di θi = ωi −∑


nPmin

Dmaxcos(ϕmax) > max

i ,j

(ωi

Di−

ωj

Dj

)+ max

i

∑j

Pij

Disin(ϕij)

1) phase cohesiveness: arc-invariance for all arc-lengths

arcsin(cos(ϕmax)

RHS

LHS

)︸︷︷︸

γmin

≤ γ ≤ π

2− ϕmax︸︷︷︸γmax

practical phase sync: in finite time, arc-length γmin

2) frequency synch: from all initial conditions in a γmax arc,exponential frequency synchronization


Main proof ideas

1 Cohesiveness θ(t) ∈ ∆(γ) ⇔ arc-length V (θ(t)) is non-increasing

V (!(t))

⇔

V (θ(t)) = max{|θi (t)− θj(t)| | i , j ∈ {1, . . . , n}}

D+V (θ(t))!≤ 0

∼ contraction property [D. Bertsekas et al. ’94, L.Moreau ’04 & ’05, Z. Lin et al. ’08, . . . ]

2 Frequency synchronization in ∆(γ) ⇔ consensus protocol in Rn

d

dtθi = −

∑j 6=i

aij(t)(θi − θj) ,

where aij(t) =Pij

Dicos(θi (t)− θj(t) + ϕij) > 0 for all t ≥ 0


Synchronization of non-uniform Kuramoto: uniform

Classic (uniform) Kuramoto Model in Tn:

θi = ωi −K

n

∑j 6=i

sin(θi − θj)

Necessary and sufficient condition

(sufficiency) synchronization condition (?) reads

K > maxi ,j

(ωi − ωj

)also necessary when considering all distributions of ω ∈ [ωmin, ωmax]

Condition (?) strictly improves existing bounds on Kuramoto model:[F. de Smet et al. ’07, N. Chopra et al. ’09, G. Schmidt et al. ’09,

A. Jadbabaie et al. ’04, S.J. Chung et al. ’10, J.L. van Hemmen et al. ’93].

Necessary condition synchronization: K > n2(n−1)(ωmax − ωmin)

[J.L. van Hemmen et al. ’93, A. Jadbabaie et al. ’04, N. Chopra et al. ’09]


Synchronization of non-uniform Kuramoto: alternative

Non-uniform Kuramoto Model in Tn - rewritten:

Di θi = ωi −∑


λ2(L(Pij cos(ϕij)))︸︷︷︸lossless connectivity

> f (Di )︸︷︷︸non-uniform Di s

·(1/ cos(ϕmax)

)︸︷︷︸phase shifts

×

×( ∣∣∣∣∣∣[ . . . ,

ωi

Di−

ωj

Dj, . . .

]∣∣∣∣∣∣2︸︷︷︸

non-uniformity

+√

λmax(L)∣∣∣∣∣∣[ . . . ,

∑j

Pij

Disin(ϕij), . . .

]∣∣∣∣∣∣2︸︷︷︸

lossy coupling

)

Similar synch, quadratic Lyap, uniform test K > ||[. . . , ωi − ωj , . . . ]||2Dorfler and Bullo (UCSB) Synchronization & Kron Reduction NECSYS ’10 @ Annecy 23 / 30

Synchronization of non-uniform Kuramoto

Non-uniform Kuramoto Model in Tn

Di θi = ωi −∑


Further interesting results:

1 explicit synchronization frequency

2 exponential rate of frequency synchronization

3 conditions for phase synchronization

4 results for general non-complete graphs

. . . to be found in our papers.


Outline





5 Conclusions


Network-preserving power network models

So far we considered a network-reduced power system model:

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synchronization conditions on λ2(P) and Pmin

all-to-all reduced admittance matrix Yreduced ∼ P/V 2

(for uniform voltage levels |Vi | = V )

Topological non-reduced network-preserving power system model:

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3320

5

34

10

3

32

6

2

31

1

8

7

5

4

3

18

17

26

2728

24

21

16

1514

13

12

11

1

39

9

topological bus admittance matrix Ynetwork

indicating transmission lines and loads (self-loops)

Schur complement:Yreduced = Ynetwork/Yinterior

c.f. “Kron reduction”, “Dirichlet-to-Neumann map”,“Schur contraction”, “Gaussian elimination”, . . .


Kron reduction of graphs: definition

Kron reduction of a graph with

boundary �� , interior •◦ , non-neg self-loops

loopy Laplacian matrix Ynetwork

25

8

27 28

8

30

30

8

30

28

1 Iterative 1-dim Kron reduction:

Topological evolution of the corresponding graph

...7)

...

8

8 8

25

8

27 28

8

30

30

8

30

28

25

8

27 28

8

30

30

8

30

X

1)

25

8

27 28

8

30

30

8

30

25

8

27

8

30

30

8

30

X

2)

25

8

27

8

30

30

8

30

25

8

8

30

30

8

30

3)

X

Algebraic evolution of Laplacian matrix: Yk+1reduced = Yk

reduced/ •◦Dorfler and Bullo (UCSB) Synchronization & Kron Reduction NECSYS ’10 @ Annecy 26 / 30

Kron reduction of graphs: properties

1 Well-posedness: set of loopy Laplacian matrices is closed

2 Equivalence: iterative 1-dim reduction = 1-step reduction

2

1030

25

8

37

29

93

8

23

7

3 6

22

635

19

4

33

20

5

34

10

3

32

6

2

31

1

8

7

5

4

318

17

26

27 2

8

24

21

16

15

14

13

12

11

1 3 9


2

30

25

37

29

38

23 3 6

22

35

19

33

2034

10

32

631

1

8

7

5

4

318

17

26

27 2

8

24

21

16

15

14

13

12

11

3 9

9

10

9

7

6

4

5

3

2

1

8


interior network connected ⇒ reduced network complete

at least one node in interior network features a self-loop

⇒ all nodes in reduced network feature self-loops

4 Algebraic properties: self-loops in interior network




Kron reduction of graphs: properties

Some properties of the Kron reduction process:...

5 Spectral properties:

interlacing property: λi (Ynetwork) ≤ λi (Yreduced) ≤ λi+n−|�|(Ynetwork)

algebraic connectivity λ2 is non-decreasing along Kron

6 Effective resistance:

Effective resistance R(i , j) among boundary nodes �� is invariant

For boundary nodes �� : effective resistance R(i , j) uniform⇔ coupling Yreduced(i , j) uniform ⇔ 1/R(i , j) = n

2 |Yreduced(i , j)|


Synchronization in network-preserving models

Assumption I: lossless network and uniform voltage levels V at generators

1 Spectral condition for synchronization: λ2(P) ≥ ... becomes

λ2(i · Lnetwork) >∣∣∣∣∣∣(ω2

D2− ω1

D1, . . .

)∣∣∣∣∣∣2· f (Di )

V 2+ min{ }

Assumption II: effective resistance R among generator nodes is uniform

2 Resistance-based condition for synchronization: nPmin≥ ... becomes

1

R> max

i ,j

{ωi

Di−

ωj

Dj

}·Dmax

2V 2


Outline





5 Conclusions


Conclusions


Time-varyingConsensus Protocols

Non-uniform Kuramoto Oscillators

singular perturbationsand graph theory

Kuramoto, consensus,and nonlinear control tools

Ambitious workplan1 sharpest conditions for most realistic models2 stochastic instead of worst-case analysis3 networks of DC/AC power inverters4 control via voltage regulation and “flexible AC transmission systems”5 “distance to instability” and optimal islanding for failure management

transition to DOE laboratories and utilities


summary - ucsbmotion.me.ucsb.edu/talks/2010u-synchkron-annecy-13sep10-2x2.pdf · er and bullo...

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